# Integral and measure-turnpike properties for infinite-dimensional optimal control systems

arXiv: https://arxiv.org/abs/1705.02762 · HTML: https://mlsanity.com/paper/1705.02762/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 May 2017.

## Built on by (most cited first, 2 of 2)

- [Turnpike in optimal control of PDEs, ResNets, and beyond](https://mlsanity.com/paper/2202.04097/) (2022)
- [Tuning the burn-in phase in training recurrent neural networks improves their performance](https://mlsanity.com/paper/2602.10911/) (2026)

## Abstract (arXiv)

We first derive a general integral-turnpike property around a set for infinite-dimensional non-autonomous optimal control problems with any possible terminal state constraints, under some appropriate assumptions. Roughly speaking, the integral-turnpike property means that the time average of the distance from any optimal trajectory to the turnpike set con- verges to zero, as the time horizon tends to infinity. Then, we establish the measure-turnpike property for strictly dissipative optimal control systems, with state and control constraints. The measure-turnpike property, which is slightly stronger than the integral-turnpike property, means that any optimal (state and control) solution remains essentially, along the time frame, close to an optimal solution of an associated static optimal control problem, except along a subset of times that is of small relative Lebesgue measure as the time horizon is large. Next, we prove that strict strong duality, which is a classical notion in optimization, implies strict dissipativity, and measure-turnpike. Finally, we conclude the paper with several comments and open problems.